Quadratic roots, full range

Percentage Accurate: 53.3% → 83.0%

Time: 10.6s

Alternatives: 7

Speedup: 19.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 53.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e-54)
   (- (/ c b) (/ b a))
   (if (<= b 3.5e-65)
     (/ -1.0 (* a (/ 2.0 (- b (hypot (sqrt (* c (* a -4.0))) b)))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.5e-65) {
		tmp = -1.0 / (a * (2.0 / (b - hypot(sqrt((c * (a * -4.0))), b))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.5e-65) {
		tmp = -1.0 / (a * (2.0 / (b - Math.hypot(Math.sqrt((c * (a * -4.0))), b))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e-54:
		tmp = (c / b) - (b / a)
	elif b <= 3.5e-65:
		tmp = -1.0 / (a * (2.0 / (b - math.hypot(math.sqrt((c * (a * -4.0))), b))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-54)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.5e-65)
		tmp = Float64(-1.0 / Float64(a * Float64(2.0 / Float64(b - hypot(sqrt(Float64(c * Float64(a * -4.0))), b)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e-54)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.5e-65)
		tmp = -1.0 / (a * (2.0 / (b - hypot(sqrt((c * (a * -4.0))), b))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-54], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.5e-65], N[(-1.0 / N[(a * N[(2.0 / N[(b - N[Sqrt[N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5000000000000005e-54

   1. Initial program 68.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 68.4%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 68.4%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 68.4%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 68.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 68.1%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 68.1%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 68.1%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 68.1%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 68.1%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around -inf 94.2%

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. mul-1-neg 94.2%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      2. unsub-neg 94.2%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   6. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -7.5000000000000005e-54 < b < 3.50000000000000005e-65

   1. Initial program 76.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 76.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 76.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 76.6%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 76.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 76.5%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 76.5%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 76.5%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 76.5%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 76.5%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 76.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ 76.6%

         \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}{a}} \]

      2. clear-num 76.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}}} \]

   5. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 76.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot a}}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}} \]

      2. times-frac 76.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \cdot \frac{a}{-0.5}}} \]

      3. fma-udef 76.4%

         \[\leadsto \frac{1}{\frac{1}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} \cdot \frac{a}{-0.5}} \]

      4. add-sqr-sqrt 76.4%

         \[\leadsto \frac{1}{\frac{1}{b - \sqrt{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}} + b \cdot b}} \cdot \frac{a}{-0.5}} \]

      5. hypot-def 79.1%

         \[\leadsto \frac{1}{\frac{1}{b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}} \cdot \frac{a}{-0.5}} \]

   7. Applied egg-rr 79.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot \frac{a}{-0.5}}} \]
   8. Step-by-step derivation

      1. frac-2neg 79.1%

         \[\leadsto \color{blue}{\frac{-1}{-\frac{1}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot \frac{a}{-0.5}}} \]

      2. metadata-eval 79.1%

         \[\leadsto \frac{\color{blue}{-1}}{-\frac{1}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot \frac{a}{-0.5}} \]

      3. div-inv 79.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{-\frac{1}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot \frac{a}{-0.5}}} \]

      4. *-commutative 79.1%

         \[\leadsto -1 \cdot \frac{1}{-\color{blue}{\frac{a}{-0.5} \cdot \frac{1}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}}} \]

      5. div-inv 79.2%

         \[\leadsto -1 \cdot \frac{1}{-\color{blue}{\frac{\frac{a}{-0.5}}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}}} \]

      6. div-inv 79.2%

         \[\leadsto -1 \cdot \frac{1}{-\frac{\color{blue}{a \cdot \frac{1}{-0.5}}}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}} \]

      7. metadata-eval 79.2%

         \[\leadsto -1 \cdot \frac{1}{-\frac{a \cdot \color{blue}{-2}}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}} \]

      8. associate-*r* 79.2%

         \[\leadsto -1 \cdot \frac{1}{-\frac{a \cdot -2}{b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}, b\right)}} \]

   9. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{-\frac{a \cdot -2}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 79.2%

          \[\leadsto \color{blue}{\frac{-1 \cdot 1}{-\frac{a \cdot -2}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)}}} \]

       2. metadata-eval 79.2%

          \[\leadsto \frac{\color{blue}{-1}}{-\frac{a \cdot -2}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)}} \]

       3. distribute-neg-frac 79.2%

          \[\leadsto \frac{-1}{\color{blue}{\frac{-a \cdot -2}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)}}} \]

       4. distribute-rgt-neg-in 79.2%

          \[\leadsto \frac{-1}{\frac{\color{blue}{a \cdot \left(--2\right)}}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)}} \]

       5. metadata-eval 79.2%

          \[\leadsto \frac{-1}{\frac{a \cdot \color{blue}{2}}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)}} \]

       6. *-lft-identity 79.2%

          \[\leadsto \frac{-1}{\frac{a \cdot 2}{\color{blue}{1 \cdot \left(b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)\right)}}} \]

       7. times-frac 79.1%

          \[\leadsto \frac{-1}{\color{blue}{\frac{a}{1} \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)}}} \]

       8. /-rgt-identity 79.1%

          \[\leadsto \frac{-1}{\color{blue}{a} \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot -4}, b\right)}} \]

       9. rem-square-sqrt 0.0%

          \[\leadsto \frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{\left(a \cdot c\right) \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}, b\right)}} \]

       10. *-commutative 0.0%

           \[\leadsto \frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)}, b\right)}} \]

       11. unpow2 0.0%

           \[\leadsto \frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{\left(c \cdot a\right) \cdot \color{blue}{{\left(\sqrt{-4}\right)}^{2}}}, b\right)}} \]

       12. associate-*r* 0.0%

           \[\leadsto \frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{\color{blue}{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}, b\right)}} \]

       13. unpow2 0.0%

           \[\leadsto \frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}\right)}, b\right)}} \]

       14. rem-square-sqrt 79.1%

           \[\leadsto \frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)}, b\right)}} \]

   11. Simplified 79.1%

       \[\leadsto \color{blue}{\frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}}} \]

   if 3.50000000000000005e-65 < b

   1. Initial program 14.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 14.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 14.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 14.6%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 14.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 14.5%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 14.5%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 14.5%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 14.5%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 14.5%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 14.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 87.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 87.8%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 87.8%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   6. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1}{a \cdot \frac{2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.65e-6)
   (/ (- b) a)
   (if (<= b 8.8e-63)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.65e-6) {
		tmp = -b / a;
	} else if (b <= 8.8e-63) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.65d-6)) then
        tmp = -b / a
    else if (b <= 8.8d-63) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.65e-6) {
		tmp = -b / a;
	} else if (b <= 8.8e-63) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.65e-6:
		tmp = -b / a
	elif b <= 8.8e-63:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.65e-6)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8.8e-63)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.65e-6)
		tmp = -b / a;
	elseif (b <= 8.8e-63)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.65e-6], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8.8e-63], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.65000000000000021e-6

   1. Initial program 65.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 65.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 65.9%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 65.9%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 65.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 65.6%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 65.6%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 65.6%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 65.6%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 65.6%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around -inf 96.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. associate-*r/ 96.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg 96.1%

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   6. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -3.65000000000000021e-6 < b < 8.7999999999999998e-63

   1. Initial program 77.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   if 8.7999999999999998e-63 < b

   1. Initial program 14.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 14.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 14.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 14.6%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 14.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 14.5%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 14.5%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 14.5%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 14.5%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 14.5%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 14.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 87.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 87.8%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 87.8%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   6. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-58}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e-58)
   (- (/ c b) (/ b a))
   (if (<= b 2.9e-65)
     (* -0.5 (/ (- b (sqrt (* a (* c -4.0)))) a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-58) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.9e-65) {
		tmp = -0.5 * ((b - sqrt((a * (c * -4.0)))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.02d-58)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.9d-65) then
        tmp = (-0.5d0) * ((b - sqrt((a * (c * (-4.0d0))))) / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-58) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.9e-65) {
		tmp = -0.5 * ((b - Math.sqrt((a * (c * -4.0)))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.02e-58:
		tmp = (c / b) - (b / a)
	elif b <= 2.9e-65:
		tmp = -0.5 * ((b - math.sqrt((a * (c * -4.0)))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e-58)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.9e-65)
		tmp = Float64(-0.5 * Float64(Float64(b - sqrt(Float64(a * Float64(c * -4.0)))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.02e-58)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.9e-65)
		tmp = -0.5 * ((b - sqrt((a * (c * -4.0)))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e-58], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-65], N[(-0.5 * N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0199999999999999e-58

   1. Initial program 68.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 68.4%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 68.4%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 68.4%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 68.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 68.1%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 68.1%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 68.1%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 68.1%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 68.1%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 68.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around -inf 94.2%

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. mul-1-neg 94.2%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      2. unsub-neg 94.2%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   6. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -1.0199999999999999e-58 < b < 2.8999999999999998e-65

   1. Initial program 76.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 76.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 76.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 76.6%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 76.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 76.5%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 76.5%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 76.5%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 76.5%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 76.5%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 76.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 76.5%

         \[\leadsto \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]

      2. *-commutative 76.5%

         \[\leadsto \left(b - \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]

      3. associate-*r* 76.5%

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]

      4. metadata-eval 76.5%

         \[\leadsto \left(b - \sqrt{\left(a \cdot \color{blue}{\left(-4\right)}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]

      5. distribute-rgt-neg-in 76.5%

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-a \cdot 4\right)} \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]

      6. *-commutative 76.5%

         \[\leadsto \left(b - \sqrt{\left(-\color{blue}{4 \cdot a}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]

      7. distribute-lft-neg-in 76.5%

         \[\leadsto \left(b - \sqrt{\color{blue}{\left(-\left(4 \cdot a\right) \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]

      8. +-commutative 76.5%

         \[\leadsto \left(b - \sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]

      9. sub-neg 76.5%

         \[\leadsto \left(b - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \frac{-0.5}{a} \]

      10. add-sqr-sqrt 76.2%

          \[\leadsto \left(b - \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\right) \cdot \frac{-0.5}{a} \]

      11. pow2 76.2%

          \[\leadsto \left(b - \color{blue}{{\left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{2}}\right) \cdot \frac{-0.5}{a} \]

   5. Applied egg-rr 76.2%

      \[\leadsto \left(b - \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}}\right) \cdot \frac{-0.5}{a} \]

   6. Taylor expanded in a around inf 41.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b - {\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{1}{a}\right) + \log \left(-4 \cdot c\right)\right)}\right)}^{2}}{a}} \]
   7. Step-by-step derivation

   8. Simplified 72.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a}} \]

   if 2.8999999999999998e-65 < b

   1. Initial program 14.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 14.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 14.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 14.6%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 14.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 14.5%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 14.5%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 14.5%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 14.5%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 14.5%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 14.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 87.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 87.8%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 87.8%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   6. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-58}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 67.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 71.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 71.7%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 71.7%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 71.7%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 71.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 71.5%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 71.5%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 71.5%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 71.5%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 71.5%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around -inf 67.3%

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. mul-1-neg 67.3%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      2. unsub-neg 67.3%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   6. Simplified 67.3%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -1.999999999999994e-310 < b

   1. Initial program 31.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 31.7%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 31.7%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 31.7%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 31.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 31.6%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 31.6%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 31.6%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 31.6%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 31.6%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 31.6%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 68.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 68.9%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 68.9%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   6. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 43.3% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-16}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.75e-16) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.75e-16) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.75d-16) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.75e-16) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.75e-16:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.75e-16)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.75e-16)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.75e-16], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.75000000000000009e-16

   1. Initial program 70.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 70.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 70.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 70.8%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 70.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 70.6%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 70.6%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 70.6%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 70.6%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 70.6%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around -inf 50.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. associate-*r/ 50.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg 50.1%

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   6. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if 1.75000000000000009e-16 < b

   1. Initial program 11.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 11.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 11.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 11.2%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 11.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 11.2%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 11.2%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 11.2%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 11.2%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 11.2%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 11.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 70.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{-0.5}{a} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 63.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \frac{c \cdot a}{b}\right) \cdot \frac{-0.5}{a}\right)\right)} \]

      2. expm1-udef 28.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot \frac{c \cdot a}{b}\right) \cdot \frac{-0.5}{a}\right)} - 1} \]

      3. associate-*l* 28.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(\frac{c \cdot a}{b} \cdot \frac{-0.5}{a}\right)}\right)} - 1 \]

      4. *-commutative 28.0%

         \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(\frac{\color{blue}{a \cdot c}}{b} \cdot \frac{-0.5}{a}\right)\right)} - 1 \]

   6. Applied egg-rr 28.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(\frac{a \cdot c}{b} \cdot \frac{-0.5}{a}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 63.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(\frac{a \cdot c}{b} \cdot \frac{-0.5}{a}\right)\right)\right)} \]

      2. expm1-log1p 70.2%

         \[\leadsto \color{blue}{2 \cdot \left(\frac{a \cdot c}{b} \cdot \frac{-0.5}{a}\right)} \]

      3. associate-*r* 70.2%

         \[\leadsto \color{blue}{\left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{-0.5}{a}} \]

      4. associate-*r/ 70.3%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(a \cdot c\right)}{b}} \cdot \frac{-0.5}{a} \]

      5. times-frac 70.4%

         \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(a \cdot c\right)\right) \cdot -0.5}{b \cdot a}} \]

      6. *-commutative 70.4%

         \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(2 \cdot \left(a \cdot c\right)\right)}}{b \cdot a} \]

      7. associate-*r* 70.4%

         \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot 2\right) \cdot \left(a \cdot c\right)}}{b \cdot a} \]

      8. metadata-eval 70.4%

         \[\leadsto \frac{\color{blue}{-1} \cdot \left(a \cdot c\right)}{b \cdot a} \]

      9. *-commutative 70.4%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(c \cdot a\right)}}{b \cdot a} \]

      10. neg-mul-1 70.4%

          \[\leadsto \frac{\color{blue}{-c \cdot a}}{b \cdot a} \]

   8. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{-c \cdot a}{b \cdot a}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 64.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-c \cdot a}{b \cdot a}\right)\right)} \]

      2. expm1-udef 27.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-c \cdot a}{b \cdot a}\right)} - 1} \]

      3. add-sqr-sqrt 17.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-c \cdot a} \cdot \sqrt{-c \cdot a}}}{b \cdot a}\right)} - 1 \]

      4. sqrt-unprod 26.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-c \cdot a\right) \cdot \left(-c \cdot a\right)}}}{b \cdot a}\right)} - 1 \]

      5. sqr-neg 26.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}}{b \cdot a}\right)} - 1 \]

      6. sqrt-unprod 16.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{c \cdot a} \cdot \sqrt{c \cdot a}}}{b \cdot a}\right)} - 1 \]

      7. add-sqr-sqrt 26.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{c \cdot a}}{b \cdot a}\right)} - 1 \]

      8. *-commutative 26.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{a \cdot c}}{b \cdot a}\right)} - 1 \]

      9. *-commutative 26.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a \cdot c}{\color{blue}{a \cdot b}}\right)} - 1 \]

      10. times-frac 26.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{a} \cdot \frac{c}{b}}\right)} - 1 \]

   10. Applied egg-rr 26.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{a} \cdot \frac{c}{b}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 25.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{a} \cdot \frac{c}{b}\right)\right)} \]

       2. expm1-log1p 25.7%

          \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{c}{b}} \]

       3. *-inverses 25.7%

          \[\leadsto \color{blue}{1} \cdot \frac{c}{b} \]

       4. *-lft-identity 25.7%

          \[\leadsto \color{blue}{\frac{c}{b}} \]

   12. Simplified 25.7%

       \[\leadsto \color{blue}{\frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-16}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6: 67.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.2e-281) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.2e-281) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 9.2d-281) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.2e-281) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.2e-281:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.2e-281)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9.2e-281)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.2e-281], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.19999999999999956e-281

   1. Initial program 72.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 72.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 72.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 72.3%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 72.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 72.1%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 72.1%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 72.1%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 72.1%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 72.1%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around -inf 65.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   5. Step-by-step derivation

      1. associate-*r/ 65.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg 65.6%

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   6. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if 9.19999999999999956e-281 < b

   1. Initial program 30.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 30.0%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 30.0%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 30.0%

         \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      4. neg-mul-1 30.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      5. associate-*l/ 30.0%

         \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

      6. *-commutative 30.0%

         \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

      7. associate-/r* 30.0%

         \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

      8. /-rgt-identity 30.0%

         \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

      9. metadata-eval 30.0%

         \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Simplified 30.0%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf 70.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.5%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 70.5%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   6. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 10.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

public static double code(double a, double b, double c) {
	return c / b;
}

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / b;
end

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. neg-sub0 51.7%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 51.7%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 51.7%

      \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   4. neg-mul-1 51.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   5. associate-*l/ 51.5%

      \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]

   6. *-commutative 51.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]

   7. associate-/r* 51.5%

      \[\leadsto \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]

   8. /-rgt-identity 51.5%

      \[\leadsto \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]

   9. metadata-eval 51.5%

      \[\leadsto \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

3. Simplified 51.6%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 25.8%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{-0.5}{a} \]
5. Step-by-step derivation

   1. expm1-log1p-u 23.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \frac{c \cdot a}{b}\right) \cdot \frac{-0.5}{a}\right)\right)} \]

   2. expm1-udef 11.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot \frac{c \cdot a}{b}\right) \cdot \frac{-0.5}{a}\right)} - 1} \]

   3. associate-*l* 11.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(\frac{c \cdot a}{b} \cdot \frac{-0.5}{a}\right)}\right)} - 1 \]

   4. *-commutative 11.1%

      \[\leadsto e^{\mathsf{log1p}\left(2 \cdot \left(\frac{\color{blue}{a \cdot c}}{b} \cdot \frac{-0.5}{a}\right)\right)} - 1 \]

6. Applied egg-rr 11.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(\frac{a \cdot c}{b} \cdot \frac{-0.5}{a}\right)\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 23.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(\frac{a \cdot c}{b} \cdot \frac{-0.5}{a}\right)\right)\right)} \]

   2. expm1-log1p 25.8%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{a \cdot c}{b} \cdot \frac{-0.5}{a}\right)} \]

   3. associate-*r* 25.8%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{-0.5}{a}} \]

   4. associate-*r/ 25.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(a \cdot c\right)}{b}} \cdot \frac{-0.5}{a} \]

   5. times-frac 25.7%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(a \cdot c\right)\right) \cdot -0.5}{b \cdot a}} \]

   6. *-commutative 25.7%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(2 \cdot \left(a \cdot c\right)\right)}}{b \cdot a} \]

   7. associate-*r* 25.7%

      \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot 2\right) \cdot \left(a \cdot c\right)}}{b \cdot a} \]

   8. metadata-eval 25.7%

      \[\leadsto \frac{\color{blue}{-1} \cdot \left(a \cdot c\right)}{b \cdot a} \]

   9. *-commutative 25.7%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(c \cdot a\right)}}{b \cdot a} \]

   10. neg-mul-1 25.7%

       \[\leadsto \frac{\color{blue}{-c \cdot a}}{b \cdot a} \]

8. Simplified 25.7%

   \[\leadsto \color{blue}{\frac{-c \cdot a}{b \cdot a}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 23.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-c \cdot a}{b \cdot a}\right)\right)} \]

   2. expm1-udef 10.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-c \cdot a}{b \cdot a}\right)} - 1} \]

   3. add-sqr-sqrt 6.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-c \cdot a} \cdot \sqrt{-c \cdot a}}}{b \cdot a}\right)} - 1 \]

   4. sqrt-unprod 9.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-c \cdot a\right) \cdot \left(-c \cdot a\right)}}}{b \cdot a}\right)} - 1 \]

   5. sqr-neg 9.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}}{b \cdot a}\right)} - 1 \]

   6. sqrt-unprod 5.8%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{c \cdot a} \cdot \sqrt{c \cdot a}}}{b \cdot a}\right)} - 1 \]

   7. add-sqr-sqrt 9.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{c \cdot a}}{b \cdot a}\right)} - 1 \]

   8. *-commutative 9.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{a \cdot c}}{b \cdot a}\right)} - 1 \]

   9. *-commutative 9.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{a \cdot c}{\color{blue}{a \cdot b}}\right)} - 1 \]

   10. times-frac 10.1%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{a} \cdot \frac{c}{b}}\right)} - 1 \]

10. Applied egg-rr 10.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{a} \cdot \frac{c}{b}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 9.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{a} \cdot \frac{c}{b}\right)\right)} \]

    2. expm1-log1p 10.2%

       \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{c}{b}} \]

    3. *-inverses 10.2%

       \[\leadsto \color{blue}{1} \cdot \frac{c}{b} \]

    4. *-lft-identity 10.2%

       \[\leadsto \color{blue}{\frac{c}{b}} \]

12. Simplified 10.2%

    \[\leadsto \color{blue}{\frac{c}{b}} \]

13. Final simplification 10.2%

    \[\leadsto \frac{c}{b} \]

Reproduce

herbie shell --seed 2023252 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))